The goal of this document is to calculate the growth rate (r) and the carrying capacity (K) of 6 species of ciliates that we grew individually in a replicated (n = 3) factorial experiment with 5 temperatures (18, 21, 24, 26, and 28), 5 nutrients levels and their combinations (= 25 treatments) for 3 weeks. The calculated r and K are then going to be used as the traits to calculate potential response diversity of communities composed of 2, 3, and 4 species to all possible changes in temperature and nutrients. The calculated potential response diversity will inform us on which communities will be used in the following experimental step. We are going to calculate the growth rate as the slope of the regression of ln(Nt) where: ln is natural log and Nt is the population density at time t, during the period of exponential growth. We are going to set initially the period of exponential growth as the first 6 days, but we are going to visually check whether this choice is correct. K will be calculated as the highest population biomass for each population during the experiment.
Let’s start loading the data set and creating a subset for calculating r
(#fig:time_series_spp)Time series of species densities across the treatments.
## # A tibble: 6 × 7
## species nutrients temperature sample_ID Intercept GrowthRate RSquared
## <chr> <int> <int> <chr> <dbl> <dbl> <dbl>
## 1 Colpidium 1 18 C1 5.43 -0.0285 0.0113
## 2 Colpidium 1 18 C2 5.06 0.135 0.510
## 3 Colpidium 1 18 C3 5.43 -0.00431 0.00466
## 4 Colpidium 1 21 C16 5.08 0.0609 0.166
## 5 Colpidium 1 21 C17 5.98 -0.125 0.601
## 6 Colpidium 1 21 C18 4.84 0.0102 0.0864
Visual inspection of exponential growth phase
(#fig:reg_all)Species densities across the treatments with the regression lines used to calculate the intrinsic rate of growth (r).
At least for now, we focus on the intrinsic rate of growth (r). We are going to fit response surfaces using the calculated r for all species using GAMs. Then we are also going to use r as the species’ trait to calculate potential response diversity, as well as response diversity when the direction of the environmental change is known. The rationale is that r is likely of more relevance if the environmental change of interest occurs rapidly, since r provides information on a population’s ability to rapidly bounce back after disturbance. In the upcoming experiment, we will have temperature fluctuating relatively fast, and so we decide now to focus on r.
Use GAMs to fit response surface of r and K
species temperature nutrients predicted
1 Colpidium 18 1.00 0.02209644 2 Colpidium 18 1.01 0.02847381 3 Colpidium 18 1.02 0.03485059 4 Colpidium 18 1.03 0.04122618 5 Colpidium 18 1.04 0.04759996 6 Colpidium 18 1.05 0.05397136
Create surface plots
We now check how the GAMs surfaces (r) look compared to the measured densities.
Checking predictions(#fig:surface_Dexi)Measured density values of Dexiostoma in the different treatments (a) vs fitted surface of growth rate (b).
(#fig:surface_colp)Measured density values of Colpidium in the different treatments (a) vs fitted surface of growth rate (b).
(#fig:surface_loxo)Measured density values of Loxocephalus in the different treatments (a) vs fitted surface of growth rate (b).
(#fig:surface_paramecium)Measured density values of Paramecium in the different treatments (a) vs fitted surface of growth rate (b).